An important source of uncertainty when estimating the
abundance of cold axions, is whether the U\(_{\rm PQ}\) (1)
symmetry is spontaneusly broken before or after the epoch
of inflationary expansion [1]}, [2]} that homogenizes
the very early universe. U\(_{\rm PQ}\) (1) becomes spontaneously
broken when the temperature falls below some critical value of
order \(f_a\) , during the so-called PQ phase transition. If this
phase transition occurs before inflation, the axion field gets
homogenized by the inflationary expansion, so that at the start
of the QCD phase transition the axion field oscillations begin
with almost the same amplitude everywhere. Unless this intial
amplitude happens to be zero, which would be an unexpected
accident, axions are produced in a process of “vacuum
realignment" [3]}. If the PQ phase transition
occurs after inflation, the axion field enters the QCD
phase transition with random initial amplitudes,
uncorrelated from one QCD horizon volume to the
next. In that case cold axions are produced by
vacuum realignment and by the decay of axion
strings and domain walls [4]}.
